Well, the problem is that a dedicated LR rig isn't going to be a very good hunting rifle (too heavy), and a hunting rifle won't be that great a LR gun, (sporter-weight barrel will heat up too fast in a long string, and may not be as accurate.)
That isn't necessarily true.
Let me go back to the M-70 243 I mentioned. I built it as a hunting rifle, 22 inch medium weight 1:10 twist. Regular M-70 wood stock. Again this was built for my wife who can't handle recoil (she has three rods between her shoulder blades from breaking her back). She had a Leupold 4X scope.
Along comes a precission rifle match which require a bit of physical effort that I had questions about do to my age, ticker, and COPD.
The M-70 is quite a bit lighter then my target rifles. I switched out the scope with a Weaver Tactical 3-12 X scope with Mil Dot,
I loaded 87 grn Berger's, (not to max) and found it had excellent accuracy to 1200 yards. Plus I could go 50 rounds or better without excess barrel heat ruining the accuracy.
You wouldn't shoot that much, that fast, in any match I ever heard of excluding the Infantry Trophy Match (which require service rifles).
The match I set this gun up for required no more then 2 shots at 10 targets in a 5 min time period, Then off to another stage giving the gun time to cool.
An F-class match would be 20 shots in 20 or 30 minutes, not fast enough to heat up the barrel that much.
As long as the barrel does not touch the stock as it heats up, moderately thin barrels (normal hunting contours) does not get too hot to shot something along the lines of the 243 - '06.
Lets go back to the Mosin a second. If you shot that rifle in rapid fire you'll notice the stock gets pretty dern hot. But you don't see accuracy falling off.
The Mosin is the exception to the rule. I've never seen a free floated Mosin that was an improvement over the existing configuration.
There a lot of myths about rifles and accuracy that you hear about on the Internet that just don't pan out on the range.
Accept what you read (including my advice) only after you try it yourself on the range.
To the OP, you mentioned you like the ideal of a 308............good choice, just remember the most accurate velocity of a 30 cal round is about 2200 fps. You'll probably want to go a bit faster but don't go over 2600 for the 168-175 grn match bullets. You're defeating your purpose.